The Optimal Transport (OT) problem with squared Euclidean cost consists in finding a coupling between two input measures that maximizes correlation. Consequently, the optimal coupling is often singular with respect to the Lebesgue measure. Regularizing the OT problem with an entropy term yields an approximation called entropic optimal transport. Entropic penalties steer the induced coupling toward a reference measure with desired properties. For instance, when seeking a diffuse coupling, the most popular reference measures are the Lebesgue measure and the product of the two input measures. In this work, we study the case where the reference coupling is not a product, focussing on the Gaussian case as a core paradigm. We establish a reduction of such a regularised OT problem to a matrix optimization problem, enabling us to provide a complete description of the solution, both in terms of the primal variable and the dual variables. Beyond its intrinsic interest, allowing non-product references is essential in dynamic statistical settings. As a key motivation, we address the reconstruction of trajectory dynamics from finitely many time marginals where, unlike product references, Gaussian process references produce transitions that assemble into a coherent continuous-time process.

翻译：平方欧氏代价下的最优输运问题旨在寻找两个输入测度之间的耦合，以最大化相关性。因此，最优耦合通常关于勒贝格测度是奇异的。通过熵项对最优输运问题进行正则化，可得到一种近似方法，称为熵最优输运。熵惩罚项使诱导的耦合趋向于具有期望性质的参考测度。例如，当寻求扩散耦合时，最常用的参考测度是勒贝格测度及两个输入测度的乘积。本文研究参考耦合非乘积的情形，并聚焦于高斯情形作为核心范例。我们将此类正则化最优输运问题简化为一个矩阵优化问题，从而能够完整描述解的形式，包括原始变量和对偶变量。除其内在价值外，允许非乘积参考在动态统计场景中至关重要。作为关键动机，我们探讨从有限时间边际分布重建轨迹动力学的问题。与乘积参考不同，高斯过程参考所产生的转移能整合为一致的连续时间过程。